Determination of the $\pi^-$ p scattering length from pionic hydrogen

We derive a closed, model independent, expression for the electromagnetic correction factor to a phenomenological hadronic scattering length extracted from a hydrogenic atom. It is obtained in a non-relativistic approach and in the limit of a short ranged hadronic interaction to terms of order ((alpha)**2)(log(alpha)) using an extended charge distribution. A hadronic negatively charged pion-proton scattering length of 0.0870(5), in units of inverse charged pion-mass, is deduced, leading to a pion-nucleon coupling constant from the GMO relation equals to 14.04(17).